Method for Reducing Inter-Cell Interference in Wireless OFDMA Networks

ABSTRACT

Protocols for OFDM/OFDMA/SC-FDMA based wireless networks provide adaptive inter-cell interference management without explicit spectrum or frequency planning. Base stations and mobile stations acquire information about subcarrier allocation from a handoff protocol. The mobile stations can also acquire this information using cognitive sensing. Cognitive sensing can be rewarded by the base station. Using this information, subcarriers can be allocated randomly, with blind optimization, or by joint optimization. The stations can use game theory to select among the different optimization strategies.

FIELD OF THE INVENTION

This invention relates generally to managing interference in wirelessnetworks, and more particularly to reducing inter-cell interference inwireless orthogonal frequency division multiplexing (OFDM) networks.

BACKGROUND OF THE INVENTION

OFDM, OFDMA and SC-FDMA

In orthogonal frequency-division multiplexing (OFDM), the availableradio frequency (RF) spectrum is partitioned into subcarriers that areorthogonal to each other. Due to the appealing features of OFDMtechnologies, such as its spectrum efficiency, easy implementation usingfast Fourier transformation (FFT) and effectiveness in mitigatingmultipart effects, OFDM is widely used in the design of the physicallayer (PHY) of networks.

In orthogonal frequency-division multiple access (OFDMA), thesubcarriers are grouped into sets and allocated to different mobilestations for parallel transceiving. OFDMA has been adopted in a widevariety of standards for broadband wireless communications, such as theIEEE 802.16e for both uplink and downlink, and 3GFP-LTE for downlink.

The basic uplink (UL) transmission scheme in 3GPP LTE uses asingle-carrier FDMA (SC-FDMA) with cyclic prefix (CP) to achieve uplinkinter-user orthogonality and to enable efficient frequency-domainequalization at the receiver side. This allows for a relatively highdegree of commonality with the downlink OFDM scheme such that the sameparameters, e.g., clock frequency, can be used.

Network Structure

In wireless OFDMA networks, a base station at the approximate center ofa cell communicates with mobile stations in its area of coverage. Whenmobile station exits a cell, it handed over to an adjacent base station.Base stations in the network exchange information with each other via abackbone or infrastructure.

Inter-Cell Interference

When geographically adjacent base stations allocate the same spectrum,inter-cell interference affects network throughput, especially for themobile stations (MSs) within range of more than one base station. Thisinterference also increases power consumption because colliding messageshave to be retransmitted.

Two techniques are commonly used for reducing inter-cell interferencedue to subcarrier collision. One technique uses random subcarrierallocation to decrease the probability that the mobile stations in theadjacent cells are allocated the same subcarriers.

The other technique uses cooperative subcarrier allocation. Thisrequires that the base stations exchange all subcarrier allocationinformation, even for mobile stations not subject to inter-cellinterference. This also requires that all subcarriers are available inboth cells at the time the subcarriers are allocated, which is usuallyimpossible to schedule in practice.

Game Theory

Game theory is a branch of applied mathematics widely applied ineconomics and the study of behaviors of entities that are able to makerational decisions. In game theory, the process of interactive decisionmaking is modeled as a game. The idea is to maximize one's payoff. Theoutcomes of the game are usually represented by numbers known asutilities. The utilities represent levels of satisfaction with theoutcome, i.e., the payoff.

The Nash Equilibrium in a game is a stable operation point at which noplayer of the game can achieve a higher utility by deviating alone fromthe current strategy. That is, when the strategies of other players arefixed, any single player of the game achieves best payoff when theplayer stays with the strategy selected at the Nash Equilibrium.

Cognitive Sensing

Cognitive sensing is used in wireless networks to learn transmission orreception parameters so that transceivers can communicate efficientlywithout each other without interfering with other licensed or unlicensedtransceivers. Cognitive sensing is based on an active monitoring ofseveral factors in tire external and internal radio environment, such asthe radio frequency spectrum, and transceiver and network state.

SUMMARY OF THE INVENTION

The invention adaptively reduces subcarrier collisions in OFDMA wirelessnetworks. The embodiments of the invention identify mobile stations(MSs) that are subject to inter-cell interference.

The base can acquire information about subcarrier allocation from thehandoff protocol. Although the BSs communicate with each other via theinfrastructure, no pre-determined action is enforced on the BS, therebyavoiding non-adaptive subcarrier allocation.

The MSs can also acquire this information using cognitive sensing.Cognitive sensing can be rewarded by the BS. The reward gives the MSs anincentive to perform the otherwise power consumptive cognitive sensing.

Using this information, subcarriers can be allocated randomly, withblind optimization, or by joint optimization. The stations can use gametheory to select among the three different strategies. The embodimentsalso identify how to determine the periodicity for which theinterference information are acquired in an optimum manner, fordifferent traffic types such as voice and data traffic and to optimizefor different types of scheduling of a radio resource.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic of a wireless network used by embodiments of theinvention;

FIG. 2 is a schematic of the network of FIG. 1 using an infrastructureand handoff information to identify interferers according to anembodiment of the invention;

FIG. 3 is a flowchart of a method for reducing inter-cell interferenceaccording to an embodiment of the invention;

FIGS. 4-5 are block diagrams of subcarrier allocation using blindoptimization according: to an embodiment of the invention;

FIGS. 6-7 are a block diagrams of subcarrier allocation using jointoptimization according to an embodiment of the invention;

FIG. 8 is a table of outcomes a two-player strategic game for allocatingsubcarriers according to an embodiment of the invention;

FIG. 9 is pseudo code for Stackelberg leader-follower game playedbetween two base stations according to an embodiment of the invention;

FIG. 10 is a schematic of cognitive sensing according to an embodimentof the invention;

FIG. 11 is a schematic of coverage by the cognitive sensing according toan embodiment of the invention;

FIG. 12 is a schematic of coverage by the cognitive with increasedsensing range;

FIG. 13 is pseudo code of a Stackelberg leader-follower game playedbetween the MS and BS to facilitate cognitive sensing according to anembodiment of the invention;

FIG. 14 is timing diagram of current age and future subcarrier usageaccording to an embodiment of the invention; and

FIG. 15 is a time diagram of shared subcarrier allocation according toan embodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Network Topology

FIG. 1 shows a wireless OFDMA network according to an embodiment of ourinvention. The network includes base stations 101, and mobile stations102-103. The circles 104 approximate the range (R) of the base stations.The coverage area within the range is called a cell. The base stationsallocate the subcarriers within its cells. It is understood that in realworld networks, the cells need not be circular, and the coverage areamay depend on antenna configurations and environmental conditions. Weuse circles only as a reasonable approximation.

Mobile stations in a segment where cells intersect can be subject tointer-cell interference. The two cells have an overlapping radius of aR107. That is, the distance between the two base stations is 2R−αR, wherea is some constant dependent on the geographic placement of the basestations.

Frequency Reuse

The process for allocating spectrum among adjacent cells is calledfrequency reuse. When the network allocates the identical spectrum toadjacent cells, the frequency reuse factor is 1.

Traffic Load

Traffic load β is the ratio of used subcarriers S to total availablesubcarriers N, i.e., β_(i)=N_(i)/S. We consider the cases when thetraffic load is β_(i)∈[0,1].

Interference Zone

The interference zone (IZ) 105 is where inter-cell interference orsubcarrier collisions can occur. The rest of the coverage area is thenon-interference zone (nIZ) 106. In the IZ, the MSs served by BSi belongto the set Γ_(i), and the MSs served by BSj belong to the set Γ_(j). TheMSs in the set Γ_(i) are subject to inter-cell interference when thesame subcarriers are allocated by BSj to the MSs in the set Γ_(j), andvice versa.

In the example network, the fractional area of the IZ 105, with respectto the total cell coverage is

$\begin{matrix}\begin{matrix}{{f(\alpha)} = {{{A_{IZ}\left( {\alpha,R} \right)}}/\left( {\pi \; R^{2}} \right)}} \\{= {\frac{2}{\pi} \cdot {\left\lbrack {{\arccos \left( {1 - \frac{\alpha}{2}} \right)} - {\left( {1 - \frac{\alpha}{2}} \right)\sqrt{{4\alpha} - \alpha^{2}}}} \right\rbrack.}}}\end{matrix} & (1)\end{matrix}$

A size and a contour of the interference zone is a function of a handoffthreshold value as well as a RF propagation environment.

Non-Interference Zone

The MSs in the nIZ 106 belong to sets Γ^(C) _(i) and Γ_(j) ^(C).

Co-Channel Interference Management for OFDMA System

When the reuse factor is 1, the same spectrum in adjacent cells ispartitioned into the S orthogonal subcarriers. These subcarriers areallocated to different MSs using OFDMA. Interference managementaccording to the embodiments of the invention allocates differentsubcarrier to the MSs in the sets Γ_(i) and Γ_(j).

Distribution of the MSs in the Cells

If the MSs are uniformly distributed in the cells, then the locations ofthe MSs can be modeled by a two-dimensional Poisson point process, whichhas a density λ₁=M₁/(πR²), where M_(i) is the total number of MSs in thecell, and R is the radius of the cell. It should be understood that theinvention can also be worked with non-uniform distributions of MSs.

Identifying MSs in IZ with Handoff

As shown in FIG. 2, mobile networks typically perform a handoff 250 totransfer a mobile station from cell to another. A number of handovertechniques are known. To perform the handoff, the MSs can maintainactive sets 201-202. Handoff thresholds H_(Add) and H_(Delete) determinewhether or not to add a BS to the active set. If the carrier tointerference-plus-noise ratio (CINR) of the BS is less than the deletehandoff threshold H_(Delete), the MS deletes the BS from the active set.If the CINR is above the add handoff threshold H_(Add), then the BS addsthe base station to the active set. The BS currently serving the MS isthe anchor BS in the active set.

In addition, the above handoff threshold values can be decreased toinclude subcarriers or blocks of subcarriers, which do not have a largeCINR but which do have sufficient power to cause interference.Typically, these “modified handoff thresholds”, called interferencethresholds, are 10 or more dB less than the normal handoff thresholdsdescribed above. These interference threshold values are not used forhandoff purpose. Instead the interference thresholds provide informationof occupied spectrum for interference management as described herein.

In other words, the invention takes advantage of the conventionalhandoff protocol to convey the information of occupied channels andspectrum. The interference threshold values are values that can bemodified according to system requirement and used to define whether acarrier, channel or spectrum is occupied. The determination of theinterference threshold values also defines the degree of tolerance orimmunity to a RF interference in the same carrier, channel or spectrum.For the purpose of simplicity, hereon, any mention of handoff or handoffinformation would implicitly include the use of the handoff informationfor setting the interference threshold values.

In FIG. 2, the MS x_(j) is located in the IZ 105 and is served by the BSj. Therefore, the active set for the MS x_(j) is {j/(A), i} 202, and theBSj is marked as the anchor (A). If the active is sent to the BS, thenthe BS can determine whether the MS is located in the IZ, or not. Forthe example in FIG. 2, the BSj can determine that the MS x_(j) is in theIZ because the active set includes BSi. The BSj can also determine thatMS y_(j) is in the nIZ because that active set is {j(A)} 201, which onlyincludes the anchor BSj.

We use the handoff information 201-203 to reduce inter-cell interferenceaccording to an embodiment of the invention.

Information Gathering with Partial Information Exchange

Information Exchange Between BSs.

The BSs can exchange the above information via a backbone orinfrastructure 210 of FIG. 2. In a joint subcarrier optimization processdescribed below, the BSs exchange all subcarrier usage information 211.In random allocation, no subcarrier usage information is exchangedbetween BSs. In the embodiment of this invention, partial subcarrierusage information is exchanged.

Operational Scheme for Interference Avoidance with Partial SubcarrierInformation Exchange

FIG. 3 shows a method for managing inter-cell interference according toan embodiment of our invention. In phase 1, the two BSs identify 310 theMSs located in the IZ 105 and the IZ 106 from the handoff information,as described above. Upon identifying the MSs in the IZ 105. Each BSrandomly allocates 320 subcarriers to the MSs in the nIZ 106, i.e., theMSs in sets Γ_(i) ^(C) and Γ_(j) ^(C). Then, the BSs exchange 330information 211 on subcarriers that are still available via theinfrastructure. During the exchange, the set of available subcarriersN_(i) ^(A) in cell i is communicated to the BSj, and the set N_(j) ^(A)is communicated to the BSi. Other information, e.g., a historical recordof subcarrier usage, if available, can also be exchanged.

During the phase 2, each base station can select 350 to optimize 351 ornot to optimize 352. The subsequent steps are random allocation 371,blind optimization 372 or joint optimization 373 can be based on thedecision 360 of the other BS.

No Optimization

If both BSs select not 352 to optimize, then the exchanged information211 is not used, and both BSs use random allocation 371 of the availablesubcarriers.

In this case, the expected number of subcarrier collision is

E[C _(R) ]=f ²(α)β_(i)β_(j) S,   (2)

where f represents the frequencies of the subcarriers.

This indicates even when the traffic loads β in both cells arerelatively light, the number of subcarrier collisions is significant andlinearly proportional to the traffic load in each cell.

Blind Optimization

FIG. 4 shows the blind optimization 372 schematically. When the BSiselects optimization 351 while the BSj does not 352, the BSi performs‘blind’ optimization. In blind optimization, the BSi does not know theexact subcarrier allocation to the set Γ_(j) of the other BS. However,the BSi does have the available subcarriers in the cell j, i.e., N_(j)^(A), after the subcarrier information exchange. Therefore, the BSi canidentity the intersection of the subcarrier set 403 of N_(i) ^(A) 401and N_(j) ^(A) 402.

To reduce subcarrier collisions, the BSi performs the blind optimizationby allocating the subcarriers outside the intersection N_(i) ^(A)∩H_(j)^(A) first. For example, BSi can logically-number the subcarriers inN_(i) ^(A) such that the subcarriers in the intersection N_(i)^(A)∩N_(j) ^(A); are labeled logically as 1, 2, 3 . . . V_(i), whereV_(i)=|N_(i) ^(A)∩N_(j) ^(A)| is the size of the subcarrierintersection. The ordering within N_(i) ^(A)∩N_(j) ^(A) can be selectedarbitrarily. The subcarriers in N_(i) ^(A)\(N_(i) ^(A)∩N_(j) ^(A)) arelabeled logically as V_(i)+1,V_(i)+1, . . . W_(i), where W_(i)=|N_(i)^(A)| is the total number of available subcarriers in cell i. With therandom subcarrier allocation to the set Γ_(i), the base station selectsthe subcarriers using the reverse order 410, i.e., the subcarriers withhigher logical numbers are selected first. In this way, thenon-overlapping subcarriers, i.e., the subcarriers that are not subjectto possible collisions, are used first, and subcarrier collision isreduced.

Nevertheless, when the number of subcarriers needed by the MSs in theset Γ_(i) is larger than W_(i)−V_(i), the subcarriers in theintersection N_(i) ^(A)∩N_(j) ^(A) need to be used, which are subject topossible subcarrier collisions.

FIG. 5 shows an example of blind optimization where the BSi isperforming blind optimization, while the BSj randomly allocate availablesubcarriers to MSs in the set Γ_(j). After the random subcarrierallocation to the MSs in the nIZ 105, the set of available subcarriersin cell i is N_(i) ^(A)={f₁₀,f₆,f₁₄,f₉,f₁₁,f₈} 501, and the set ofavailable subcarriers in cell j is N_(j)^(A)={f₁₉,f₁₃,f₁₇,f₂₆,f₇,f₁₁,f₈} 502.

The MSs in the set Γ_(i) need four subcarriers 511, while the MSs in theset Γ_(j) need three subcarriers 512. In this example, the BSidetermines that N_(i) ^(A)∩N_(j) ^(A)={f₁₁,f₈}. Therefore, the BS labelsthe subcarrier f₈ and f₁₁ with the lowest logical numbers, and theremaining available subcarriers are labeled by logical label 3, 4, 5, 6.

In the process of allocating subcarriers, the BS prioritizes the use ofthe subcarriers with higher logical labels. Therefore, because foursubcarriers 511 are needed, BSi allocates the subcarriers{f₁₀,f₆,f₁₄,f₉} for usage in the set Γ_(i). The three subcarriers 512allocated to MSs in the set Γ_(j) are randomly selected. The BSjallocates subcarrier {f₇,f₈,f₁₃} to MSs in the set Γ_(j). Although thesubcarrier usage in the set Γ_(j) includes the subcarrier f₈∈N_(i)^(A)∩N_(j) ^(A), subcarrier collision is still reduced due to the blindoptimization performed by BSi.

The expected number of subcarrier collisions using blind optimizationperformed by BSi is

$\begin{matrix}{{{E\left\lbrack C_{B} \right\rbrack} = {M \cdot \frac{\beta_{i}{f(\alpha)}}{\left( {1 - \beta_{j}} \right) + {\beta_{i}{f(\alpha)}}}}},} & (3)\end{matrix}$

where M=l_(p)(d+1, n_(B)−d)·l_(p)(d, n_(B)−d)E[V]−dl_(p) ²(d+1,n_(B)−d), in which

${{I_{p}\left( {a,b} \right)} = \frac{\int_{0}^{\infty}{{t^{a - 1}/\left( {1 + t} \right)^{a + b}}\ {t}}}{\int_{0}^{1}{{t^{a - 1}/\left( {1 - t} \right)^{b - 1}}\ {t}}}},{\forall{\left\{ {a,b} \right\} \geq 0}}$

is a regularize beta function, and the other parameters are

E[V]=n _(B) p _(B)

n _(B) =[f(α)β_(j)+(1β_(j))]S

p _(B) =f(α)β_(i)+(1−β_(i))

d=S(1−β_(i))   (4)

The expected number of subcarrier collisions for blind optimization 372is always smaller than the number of collisions due to random allocation371, i.e., E[C_(B)]≦E[C_(R)],∀{β_(i),β_(j)}∈[0,1] and α∈[0,2].

When the traffic load is relatively light, the expected number ofsubcarrier collision is significantly lower with the blind optimization.For example, when the traffic load in both cells is smaller than 0.5.the expected number of subcarrier collisions approaches zero.

The effectiveness of the blind optimization depends on the traffic loadsin both cells as well as the number of subcarriers required by MSs inthe IZ. When the BSi performs the blind optimization, the expectednumber of subcarrier collisions monotonically decreases with β_(i).However, lower traffic load in cell j may not necessarily reduce theexpected number of subcarrier collisions. This is because as β_(j)decreases, the number of subcarriers that can potentially causeinterferences to the MSs in the set Γ_(i) ^(C) is reduced.

However, lighter traffic also means that N_(i) ^(A)∩N_(j) ^(A) is alarger set, which has negative impact on the effectiveness of blindoptimization. Therefore, the impact of lighter traffic has to beconsidered to determine whether a smaller β_(j) results in moreeffective blind optimization. Additionally, because a subcarriercollision is mutual and is equally undesirable to both the BSs, anyeffort to reduce subcarrier collision in one BS is beneficial to bothBSs.

Joint Optimization

FIG. 6 shows the joint optimization 373 after the exchange of subcarrierinformation N_(i) ^(A) and N_(j) ^(A). Both the BSs identify the setN_(i) ^(A)∩N_(j) ^(A) 403, and both BSs logically-number the availablesubcarriers such that the V overlapping available subcarriers arerespectively ordered W_(i),W_(i)−1 . . . , W_(i)−V+1 and W_(j),W_(j−)1 .. . , W_(j)−V+1, see FIG. 6. The exact ordering within N_(i) ^(A)∩N_(j)^(A) is irrelevant.

With the joint optimization, minimum subcarrier collision can beachieved given that the subcarrier usage in the nIZ is randomlyselected. That is, after subcarriers are allocated randomly in the nIZ,the joint optimization first allocates the non-overlapping subcarriersin N_(i) ^(A) to MSs in the set Γ_(i), and the non-overlappingsubcarriers in the N_(j) ^(A) to MSs first in the set Γ_(j). Both BSs donot allocate subcarriers in N_(i) ^(A)∩N_(j) ^(A), unless absolutelynecessary. In this way, when only partial information on the subcarrierusage is exchanged, both BSs make a best effort to avoid interference inthe IZ, and can further reduce the expected number of subcarriercollisions.

FIG. 7 shows an example of the joint optimization. After the randomsubcarrier allocation to the MSs in the nIZ, the set of availablesubcarriers in the cell i is N_(i) ^(A)={f₁₀,f₆,f₁₄,f₉,f₁₁,f₈}701, andthe set of available subcarriers in cell j is N_(j)^(A)={f₁₉,f₁₃,f₁₇,f₂₆,f₇,f₁₁,f₈} 702. The MSs in the set Γ_(i) needsfive subcarriers 711, while the MSs in Γ_(j) needs six subcarriers 712.In this case, both the BSi and the BSj identify the intersection N_(i)^(A)∩N_(j) ^(A)={f₁₁,f₈} 713 of the available subcarrier sets in the twocells. Therefore, both the BSi and BSj label the subcarrier f₈ and f₁₁with the highest logical numbers, specifically, f₈ and f₁₁ are logicallylabeled 6 and 5 respectively BSi, while f₁₁ and f₈ are logically labeledas 7 and 6 by BSj. The rest of the available subcarriers are logicallylabeled beginning at 1.

In the process of allocating subcarriers, both the BSs prioritize theuse of the subcarriers with lower logical labels. Therefore, becausefive subcarriers are needed, BSi allocates the subcarriers{f₁₀,f₆,f₁₄,f₉,f₁₁} for usage in Γ_(i), and BSj allocates six 6subcarriers {f₁₉,f₁₃,f₁₇,f₂₆,f₇,f₈} to MSs in Γ_(j). The subcarrierusage in the set Γ_(i) includes subcarrier f₁₁∈N_(i) ^(A)∩N_(j) ^(A) andthe subcarrier usage in Γ_(j) includes subcarrier f₈∈N_(i) ^(A)∩N_(j)^(a). However, subcarrier collision is avoided because both BSs engagein the joint optimization.

When joint optimization is perform, the expected number of subcarriercollision is

E[C _(j) ]=I _(p)(d′+1,γ_(J))·1_(p)(d′,γ _(J))E[V]−d′1_(p) ²(d′+1,γ_(J))  (5)

in which d′=S(2−β_(i)−βj) and γ_(J)=max(0,n_(B)−d′). It can be shownthat E[C_(j)]≦E[C_(B)]≦E{C_(R)],∀{β_(i),β}∈[0,1] and α∈[0,2], which,indicates a reduction in subcarrier collisions when compared to theblind optimization. Equation (5) indicates that when the traffic load inboth cells is smaller than 0.8, the joint optimization can avoid almostall subcarrier collisions.

Interactive Decision Making

The random allocation, blind optimization and joint optimization areenabled by interactions between the BSs, and cannot be solely determinedby one BSs. For example, if BSi selects to optimize, the outcome cannotbe determined until BSj has made a specific decision. If BSj does notoptimize, then the outcome is blind optimization 372 by BSi. If BSjdecides to optimize too, the outcome is joint optimization 373.

Because the decision depends on the interaction of two entities, i.e.,the base stations, game theory can be used to maximize the payback,i.e., network performance.

Strategic Game

FIG. 8 shows the outcomes of a strategic game according to an embodimentof the invention. The two players are BS 1 and BS 2. This makes our gamebase station centric. As described below, the game can also be mobilestation centric. Both players can select their action from the binarystrategy space {O, NO}, where O stands for “Optimize” 801 and NO means“Not Optimize” 802. The action of BS 1 changes row-wisely and the actionof BS 2 changes column-wisely. Different pairs of the actions taken bythe two players result in different outcomes of the game, which areassociated with different utilities (payoffs) to the two players.

An outcome of the game can be presented by a two dimensional vector Xand there are four possible game's outcomes 410 shown in Table 1.

TABLE 1 Outcome X = {action 1, action 2} Descriptions {O, O} Bothplayers coordinate in optimization to result in joint optimization. {NO,O} Player 2 optimizes while player 1 does not to result in blindoptimization performed by player 2 and player 1 gets a ‘free ride’*. {O,NO} Player 1 optimizes while player 2 does not to result in blindoptimization performed by player 1 and player 2 get a free-ride. {NO,NO} Neither of the players optimizes to result in random allocation. *Ingame theory, a free ride is gaining a benefit without the ordinaryeffort or cost.

Utility of a strategy is presented by a real number, a higher value ofutility corresponds to a higher level of satisfaction the player hasabout the outcome. The utilities of the two players in FIG. 8 arewritten in the format of [Utility of Player 1, Utility of Player 2]. Forexample, the pair {O,O} results in the outcome of joint optimization, inthis joint optimization, the players obtain utility [4,4],

Rational players try to maximize their utilities. The result of seekingmaximal utilities by different players can result in a stable outcome inwhich all players achieve their maximum utilities, under the constraintthat the actions of the other player are known. The Nash Equilibriumdescribes such a stable outcome, and is defined as the outcome in whichno player can be better off by deviating alone from the outcome.

In FIG. 8, the only Nash Equilibrium is the strategy pair {O,O}.However, when the utilities associated with different outcomes change,the Nash Equilibrium of the strategic can also change. In some cases,there can by multiple Nash Equilibria in a game. The modeling ofutilities is of vital importance to determine the stable outcome, i.e.,Nash Equilibrium, of a game.

Modeling of the Utility Function

In the strategic game shown in FIG. 8, the utilities of players aregiven directly as real numbers. However, careful modeling is required toaccurately represent the different utilities associated with differentoutcomes. In this invention, the utility of a strategy is defined as thevalue of a certain outcome to the player minus the cost required toachieve such outcome.

Most specifically, let θ be the expected number of subcarrier collisionsand θ _(X) ^(i) be the expected number of subcarrier collisions in theset Γ_(i) when the outcome is X. The value of the outcome X to BSi canthen be given as ψ(θ _(X) ^(i)), where ψ(θ) is a monotonicallydecreasing function of θ, such that

${\frac{\partial{\psi (\theta)}}{\partial\theta} < 0},$

because a rational BS prefers an outcome with reduced subcarriercollisions.

In addition, the process of optimization can be associated with a cost.Therefore, different outcomes are associated with different costs. Thecost reflects the negative impact of performing optimization. Forexample, optimization may not be performed in real time. This can causea delay in subcarrier allocation.

The cost incurred by the BSi to achieve the outcome X is a non-negativefunction g _(X) ^(i)( X). The value of the function g _(X) ^(i)( X)depends on the specific outcome X, as well as the complexity ofperforming different types optimizations, e.g., joint optimization orblind optimization. The rule of thumb for modeling the cost function isthat the cost is higher when the BSi is involved in joint optimization,compared to the case when the BS performs blind optimization. Whenrandom subcarrier allocation is selected by BSi, the cost is 0.Therefore, the utility of the BSi in outcome X is

U _(X) ^(i)=ψ(θ _(X) ^(i))−g _(X) ^(i)( X ).   (6)

With this function, if the cost of performing optimization is higher,then the outcomes of the joint optimization and blind optimization areless likely. Conversely, if the payoffs increase more rapidly with thereduction in expected number of subcarrier collisions, i.e., even asmall reduction in the number of collided subcarriers is viewed by theBS to be very valuable, then the outcome of joint optimization and/orblind optimization is more likely.

The functionality of the value function ψ(θ) and cost function g _(X)^(i)( X) reflects how the BSs evaluate the outcome and the cost ofoptimization. These functions can be chosen differently for differentapplications, under the general guidelines described above. As long asthe function ψ(θ) is monotonically decreasing, and g _(X) ^(i)( X) isnon-negative, then the Nash Equilibrium can be found by the two BSs.

Stable Outcome from the Interactions Between BSs

The outcome of the game, e.g., the four different strategy pairs in FIG.8 and Table 1 result from the interactions between two BSs. Neither ofthe two BSs can independently determine the outcome. We model theinteractions between the two base stations as a binary Stackelbergleader-follower game. The Stackelberg leadership model is a strategicgame where a leader decides first, and then the followers decidesequentially.

FIG. 9 shows the rules of this strategic game in the form of pseudocode. After exchanging the partial information on the availablesubcarriers, one BSi is the leader, while the other BSj is follower. Theterm leader does not indicate any superiority over the follower.Instead, the term strictly describes the order of sequential moves madeby different players in the game. There are no guaranteed advantages ordisadvantages associated with being a leader or a follower.

To maximize payoff (utility), the BSi first predicts the rational(optimal) response of the BSj to different actions of BSi (statement:(i) 901). In this way, the BSi can determines the different outcomesassociated with its own different actions, under the rationalityassumption of BSj. From these predicted outcomes, BSi can then selectits own optimal decision that leads to the optimal outcome A_(i)*(statement (ii) 902).

After the BSi makes its decision, the BSj observes the action of theBSi. The BSj is now fully aware of the outcomes from different actionsof its own, and if the BSj is rational, the BSi makes the optimaldecision predicted earlier by the BSi (statement (iii) 902). This leadsto the Nash Equilibrium in which the outcome is [A_(i)*,A_(j)*]. Thisrational decision, making is based on the utility modeled describedabove. The strategic game is adaptive to the different parameters of thesystem, e.g., the traffic load and a size of the interference zone.

Information Gathering Through MSs

Cognitive Sensing by MSs

When independent BSs do not communicate with each other and a MS cannotcommunicate directly with the BS of another cell, the information onpossible interferers can only be gathered from other MSs. For example,the information collecting is initiated in cell i. The MSs in the setΓ_(i) can perform cognitive sensing. In cognitive sensing generally,

transceivers learn and adapt to the environment in which they operate,in one embodiment of the invention, possible interferers are identified.

FIG. 10 shows cognitive sensing of MSs according to an embodiment of theinvention. To identify the possible interferers, a MS l 1001 in the setΓ_(i) sends out a probing signal l, which covers a sensing region A_(l)1002 of radius r_(l) 2003. If the MS 1004 in the set Γ_(j) (active orinactive) is located in the region A_(l), i.e., in the range of r_(l)from MS l, The MS 1004 responds to the probing signal of l, and reportthe subcarrier(s) that it is currently using.

Spontaneous Reliable Response

If the MS in the set Γ_(i) sends out the probing signal, all the MSs inthe set Γ_(j) that receive the probing signal respond to the probingrequest. This model is based on the rationale of mutual collision, i.e.,a co-channel subcarrier collision is equally detrimental to all MSsinvolved. Therefore, whenever the MS in the set Γ_(j) receives theprobing, it reports its presence and subcarrier usage to facilitate thepossible optimization performed by the BSi and avoid mutual subcarriercollisions. However, some MSs 1005 that are out of range of the probingsignal may go undetected.

Completeness of the Collected Information by MSs

As shown in FIG. 11, the accuracy on the subcarrier usage acquired byBSi as a result of the cognitive sensing of MSs in the set Γ_(i) dependson the fraction of area in the IZ that is covered by the cognitivesensing performed distributively at different MSs in the set Γ_(i). Thefraction of uncovered area 1101 corresponds to the fraction of blankregion in FIG. 11. For example, if 80% of the IZ is covered by thecognitive sensing of some MSs in Γ_(i), the BSi can acquire 80% of thesubcarrier used by Γ_(j) in IZ. In the example shown in FIG. 11, two MSs1105 in the set Γ_(j) are located in the blank region 1101 and do notfall within the sensing range of any MS in Γ_(j). Therefore, thepresence of these two MSs, and the subcarrier used by the two MSs cannotbe collected and reported to the BSi.

With a uniform distribution of MSs, the completeness of the informationcollected on current subcarrier usage is equivalent to the fraction ofarea covered by the MSs in Γ_(i). Because the interference is onlypossible within IZ, the completeness of the current subcarrier usageinformation is

$\begin{matrix}{{{E\left\lbrack p_{\mu} \right\rbrack} = {\mathrm{\Upsilon}_{A} = \frac{{\left( {A_{1}\bigcup{A_{2}\mspace{11mu} \ldots}\;\bigcup{A_{\mspace{11mu}}\ldots}} \right)\bigcap A_{IZ}}}{A_{IZ}}}},} & (7)\end{matrix}$

where A_(IZ) denotes the IZ, A_(f) is the sensing region of MS l,(lΕΓ_(i)), and |A_(IZ)| denotes the area of the IZ. The valuep_(μ)∈[0,1] is the extent of completeness of the collected information,and γ_(A) is the fraction of area in A_(IZ) covered by the cognitivesensing. Therefore, we have E[p_(C)]=γ_(A). When the distribution of theMSs is uniform, e.g., a two-dimensional Poisson distribution, the valuesγ_(A) and p_(μ) depend on the sensing radius r_(l). In our embodiments,the sensing radiuses of all MSs in the set Γ_(i) are modeled to be thesame.

In the example network shown in FIG. 11, the cognitive regions are shownby the smaller circles each with radius r_(l). The circles can overlapeach other. While most of the area in IZ is within one or more sensingcircles, two MSs 1105 in Γ_(j) do not fall within any cognitive sensingcircle.

As shown in FIG. 12, the blank region in the IZ can is reduced as thesensing radius r_(l) 1201 increases. In FIG. 12, the sensing radius 1201is increased such that each and every MS in the set Γ_(j) is covered bythe cognitive sensing. In the this case, the complete information on thesubcarrier usage can be acquired at BSi. In the example shown in FIG.12, there is still some blank uncovered region in the IZ, however, eachMS in Γ_(j) is within at least one of the sensing circles and completesubcarrier usage information can be acquired.

Critical Sensing Radius and Critical Ratio

The completeness of the subcarrier usage information depends on thesensing radius r_(l). Therefore, the choice of the radius r_(l) by theMSs in the set Γ_(i) has an impact on how well the BSi can determine thesubcarrier usage from the possible interferers, i.e., MSs in the setΓ_(j). The optimal sensing radius is

|Γ_(i)|·(π{circumflex over (r)} ²)=|A _(IZ) |=f(α)πR ²,   (8)

where the area f(a) of the IZ is define in Equation (1). The radius{circumflex over (r)} is

$\begin{matrix}{\hat{r} = {R{\sqrt{\frac{f(\alpha)}{\Gamma_{i}}}.}}} & (9)\end{matrix}$

The sensing radius {circumflex over (r)} provides an ideal lower-boundfor the actual sensing radius to cover the area |A_(IZ)| with |Γ_(i)|MSs with non-overlapping sensing region. Given the approximate circularshape of A_(l), it is impossible to cover a larger area withnon-overlapping sensing region; however, {circumflex over (r)} still canbe used as a lower bound.

The actual sensing radius of the MS is related to {circumflex over (r)}by

r_(l)=ε{circumflex over (r)},   (10)

where ε∈R⁺ is a critical ratio of the sensing radius. When ε∈[0,1], thecritical ratio defines the inadequacy of the actual sensing radius,because even the lower bound on the sensing radius is not yet met. Whenε∈[1,°∞), it describes the intentional redundancy of the sensing radiusto improve the probability of covering the IZ in the different sensingregions,

Completeness of Information and Sensing Radius

In a Boolean sensing model, where points are randomly placed on aninfinite plane with density λ_(D), if circles of radius of r are drawncentered at the randomly-placed points, the probability that a spot onthe infinite plane is not covered by any of these circles is

f _(u)=exp(−λ_(D) πr ²).   (11)

Although we use a finite IZ instead of an infinite plane, we can stilluse the above result to approximate the relationship between thecompleteness of information and the sensing radius r_(l), which isassumed to be identical for all the MSs in the set Γ_(i). Specifically,if the density parameter is

${\lambda_{D} = \frac{\Gamma_{i}}{A_{IZ}}},$

and the sensing radius is r_(l)=ε{circumflex over (r)}, the expected

completeness of the information collected on the subcarrier usage in IZis

$\begin{matrix}\begin{matrix}{{E\left\lbrack p_{\mu} \right\rbrack} = {1 - {\exp \left( {{- \lambda_{D}}\pi \; r_{}^{2}} \right)}}} \\{= {1 - {\exp \left\{ {{- \frac{\pi {\Gamma_{i}}}{A_{IZ}}}\left( {{ɛ \cdot R}\sqrt{\frac{f(\alpha)}{\Gamma_{i}}}} \right)^{2}} \right\}}}} \\{= {1 - {{\exp \left( {- ɛ^{2}} \right)}.}}}\end{matrix} & (12)\end{matrix}$

As the critical ratio increases, the probability that the entire IZ iscovered increases at an exponentially-squared rate. Therefore, when theMSs in the set Γ_(i) are able to increase the sensing radius, i.e., thesensing range is larger than the minimal requirement {circumflex over(r)}, the subcarrier usage information in Γ_(j) that is acquired at theBSi quickly becomes complete.

Reward from BS

Because transmitting probing signal to collect the subcarrier usageconsumes power, the BS should encourage the cognitive sensing byrewarding the MSs that perform the cognitive. This reward shouldincrease monotonically with the sensing range of MS, because powerconsumption for sensing increases with larger sensing range. In oneembodiment, the reward from BS is in the form of an increase in thedownlink transmit power. Increasing the transmit power can increase therange of the MS, increase the received SNR at MS, and decrease therequired receive power for MS.

When a MS does not perform cognitive sensing and there is no reward, theMS in the IZ is allocated transmission power P₀. However, if a MS l inthe IZ collects the subcarrier usage information within a radius r_(l),the BSi allocates transmission power P₀+R_(r)r_(l) ^(ρ). In theadditional reward power R_(r)r_(l) ^(ρ),R_(r) is the reward factorselected by the BSi and ρ is a path loss exponent that usually takesvalue in a range 2 to 5, which describes how power-consuming the sensingprocess is for the MS with respect to the sensing range.

Tradeoffs for MS in Cognitive Sensing

As slated above, one cost for the MSs in the Γ, to perform the cognitivesensing is the power consumption associated with the sensing. Let C₁ bea positive constant that specifies the power required to perform thecognitive sensing within a circular region of unit radius. The cost ofpower consumed for the sensing within he radius r_(l) can then be givenas P_(S)=C₁r_(l) ^(ρ). The path loss exponent ρ describes how the powerconsumption in sensing varies with sensing radius, and depends on thewireless environments of the MS. The value of ρ usually is in the rangeof [2, 5], where ρ=2 corresponds to the free-space line-of-sight (LOS)environment and ρ=5 describes a lossy indoor environment.

The benefit of performing cognitive sensing is two-folded. First, thepower reward from the BS can compensate for the power consumed by MSitself. Second, the information collected by the cognitive sensing canbe used by the BS to reduce subcarrier collisions, which wastesreceiving power in the MS.

To model the two benefits of the cognitive sensing, let C′₂ be theamount of receive power wasted when a subcarrier collision occurs,p_(IC) be the probability of a subcarrier collision due to theincompleteness of the collected information, i.e., the blank area inFIG. 11. Then we have

$\begin{matrix}\begin{matrix}{p_{IC} = {\beta_{j}{f(\alpha)}\left( {1 - {E\left\lbrack p_{\mu} \right\rbrack}} \right)}} \\{= {\beta_{j}{{f(\alpha)} \cdot {{\exp \left( {- ɛ^{2}} \right)}.}}}}\end{matrix} & (13)\end{matrix}$

If C₂=C′₂β_(j)f(α), then the power wasted in receiving collided downlinktransmission is P_(W)=C₂ exp(−ε²).

The downlink transmission, power reward the MS receives from the BS isR_(r)r_(l) ^(ρ), which also serves as the incentive for the MS toperform cognitive sensing. Because such reward is beneficial to the MS,the reward is related to an equivalent saving in MS's own battery powerconsumption. This equivalency can be modeled by a translation factorσ_(i,P)∈R⁺. With the translation factor σ_(l,r), the reward R_(r)r_(l)^(ρ) by the BS on the downlink transmission power is equivalent to apower saving of

$\frac{R_{r}r_{}^{\rho}}{\sigma_{i,r}}$

at the MS.

The translation factor σ_(i,r) is a “currency exchange rate” thatfacilitates the MS to measure the value of the increase in downlinktransmission power with respect to its own power saving. The choice ofσ_(i,r) is related to how MS compares the benefits of increasingdownlink power with decreasing sensing power consumption. For example,when σ_(l,r)>1, the factor indicates that the increase in downlinktransmission power is less valuable than the saving in its own batterypower, while σ_(l,r)<1 means that the MS prefers an increase in downlinktransmission power.

Therefore, the total power saving for MS that performs cognitive sensingwithin radius of r_(l) is

$\begin{matrix}{P_{MS} = {{\left( {\frac{R_{r}}{\sigma_{t,r}} - C_{1}} \right)\left( {ɛ\; \hat{r}} \right)^{\rho}} - {C_{2}{{\exp \left( {- ɛ^{2}} \right)}.}}}} & (14)\end{matrix}$

Tradeoffs for Encouraging Cognitive Sensing

The downlink transmission power of the BS is usually also limited byregulations. Therefore, the BS is also interested in minimizing itsdownlink transmission power while providing services to the MSs.

To motivate the MSs in Γ_(i) to engage in collecting information ofsubcarrier usage in Γ_(j), the BSi increases the transmission power tothe MSs to encourage cognitive sensing. The return of this increase intransmission power is that the BSi can acquire the information on thepossible interferers in the Γ_(j). Therefore, the net return in powersaving to BSi by rewarding cognitive sensing of the MSs is

P_(BS)=β_(j) f(α)(P ₀−exp(−ε²)·[P₀ +R _(r)({circumflex over (r)}ε)^(ρ)]).   (15)

Continuous Stackelberg Leader-Follower Game Between BS and MSs

Modeling Utilities for BS and MS

To understand the behaviors of the BS and the MSs, their interactionscan be modeled as multiple Stackelberg leader-follower games.Specifically, to simplify the model, all the MSs in the set Γ_(i) selectthe same sensing radius r_(l). In this case, multiple Stackelberg gamescan be modeled by a single two-player game between the BS and a MS. Inthe following, the translation factor is σ_(i,r)=1.

The game is to maximize power saving. The utilities of BS and MS dependon a number of parameters, such as the power C₁ required for sensing aunit circular region. This is a direct result of hardware design of theMS unit. The amount of power wasted in subcarrier collision isC₂=C′₂β_(j)f(α), where C′₂ is the amount of receive power wasted when asubcarrier collision occurs. The critical radius determined by thedistribution of MSs and the network topology is

${\hat{r} = {R\sqrt{\frac{f(\alpha)}{\Gamma_{i}}}}},$

and ρ is the path loss exponent that is only determined by the wirelessenvironment near the MS. Therefore, none of the parameters C₁, C₂{circumflex over (r)} or ρ can be controlled by the MS or BS in thegame. Furthermore, the power P₀ does not directly affect the attitude ofeither player towards cognitive sensing. The value of the translationfactor σ_(i,r) serves to convert transmission power reward to the powersaving of the MS, and does not capture the essential aspect forunderstanding the interactions between BS and MS, therefore, it is alsoassumed pre-fixed in the game.

From the above, the parameters that can be determined by the two players(BS and MS) of the game and affect the outcome of the game are R_(r) andε. Therefore, the utility of the two players in the game can be writtenas

U _(BS)(ε,R _(r))=P ₀−exp(−ε²)·[P ₀ +R _(r)({circumflex over(r)}ε)^(ρ)]  (16)

and

$\begin{matrix}{{{U_{MS}\left( {ɛ,R_{r}} \right)} = {P_{MS} = {{\left( {\frac{R_{r}}{\sigma_{i,r}} - C_{1}} \right)\left( {ɛ\; \hat{r}} \right)^{\rho}} - {C_{2}{\exp \left( {- ɛ^{2}} \right)}}}}},} & (17)\end{matrix}$

which are derived from Equations (14) and (15), respectively.Particularly, the BS can select R_(r) while the MS selects the value ofε.

Decision Makings in the Stackelberg Game Between BS and MSs

The decision space of the BS is {R_(r):R_(r)∈R⁺}, and the decision spaceof the MS is {ε:ε∈R⁺}. That is, the decision of both players in the gamespan a continuous space R⁺. Therefore, the game is modeled as acontinuous Stackelberg game. In this Stackelberg game, the sequentialdecisions are initiated by the BSi, which acts as the leader and thedecides first. The MS is the follower, which makes a decision afterobserving the decision from the BSi.

The value of {circumflex over (r)} is determined objectively fay BSibecause {circumflex over (r)} is determined by Equation (9) and is fixedfor a given network topology. The BSi as the leader decides on the valueof the reward factor R_(r) and broadcasts R_(r) to all MSs in Γ_(i). Thefollower (MS) observes the decision of the BSi and determines thesensing: range r_(l)=ε{circumflex over (r)} by deciding on the value ofε.

FIG. 13 shows the pseudo code for the interactive decision makingprocess. Initialization determines the values of thenon-decision-related parameters, i.e., C₁, C₂, {circumflex over (r)},P_(o), ρ and σ_(i,r). Decision making between the two players beginswith the BS. To maximize its own utility, the BS has to derive theoptimal decision on R_(r). Because the utility U_(BS)(ε,R_(r)) of the BSis determined by both ε and R_(r), the BS has to predict the decision onε that is taken by MS. To do so, BS assumes rationality in the MS andpredicts that the MS always makes the optimal decision that maximizesits own utility as well. Thus, the BS can determine the optimal responseof MS {tilde over (ε)}*=g₁(R_(r)) as a function of R_(r) in statement(i) 1301.

Then, the BS can reduce its own utility function to a function, of onlyone single variable R_(r), i.e., U_(BS)*(R_(r))=U_(BS)(g₁(R_(r)),R_(r))in statement (ii) 1302, from which the BS can determine its own optimaldecision R_(r)* in statement (iii) 1303. After BS makes its decision onthe reward factor R_(r)=R_(r)*, the MS observes the decision, anddecides the range for cognitive sensing by determining the optimal valueof critical value ε*, such that the sensing radius isr_(l)=ε*{circumflex over (r)}. This range maximizes its own utilityU_(BS)(ε,R_(r)*) in statement (iv) 1304.

Optimal Strategy and Nash Equilibrium

To determine the optimal strategies adopted by different players, a pathloss exponent of ρ=2 is used. The prediction by the BS upon the rationalresponse from MS suggests that the optimal decision of the MS can bedescribed as follows.

If R_(r)≧σ_(i,r)C₁, then the MS always performs cognitive sensing with,a maximal sensing range according to its sensing capacity.

${{{If}\mspace{14mu} 0} < C_{2} \leq {\left( {C_{1} - \frac{R_{r}}{\sigma_{t,r}}} \right){\hat{r}}^{2}}},$

the MS never performs cognitive sensing.

${{{If}\mspace{14mu} 0} < {\left( {C_{1} - \frac{R_{r}}{\sigma_{t,r}}} \right){\hat{r}}^{2}} < C_{2}},$

then the optimal response of the MS, is to perform cognitive sensingwithin a range of r_(ε)={tilde over (ε)}*{circumflex over (r)}, where{tilde over (ε)}* is a function of R_(r), and can be given as

$\begin{matrix}{{{\overset{\sim}{ɛ}}^{*}\left( R_{r} \right)} = {\sqrt{\ln \; \frac{C_{2}}{\left( {C_{1} - \frac{R_{r}}{\sigma_{t,r}}} \right){\hat{r}}^{2}}}.}} & (18)\end{matrix}$

The optimal response of the MS, as described above, suggests that whenthe MS prefers the reward on the downlink transmission power and/or thereward factor is large enough compared to the cost of sensing, i.e.,R_(r)≧σ_(i,r)C₁, the power consumed by cognitive sensing always isbeneficial. In this case, the MS tries to cover a sensing region aslarge as possible.

In contrast, if the cost of sensing is large compared to the possiblereward, then the MS never performs cognitive sensing.

The two scenarios above correspond to the trivial cases when cognitivesensing is always profitable or always harmful to MS. While such extremecases may exist, usually the balance in practice is some where inbetween.

When

${0 < {\left( {C_{1} - \frac{R_{r}}{\sigma}} \right){\hat{r}}^{2}} < C_{2}},$

whether or not a specific sensing radius is profitable depends directlyon the reward factor R_(r) promised by BS, and the optimal response canbe given as a function of R_(r) in Equation (18).

The following embodiments are based on the last scenario. The utility ofthe BS when the MS selects the optimal response is

$\begin{matrix}\begin{matrix}{{U_{BS}^{*}\left( R_{r} \right)} = {U_{BS}\left( {{{\overset{\sim}{ɛ}}^{*}\left( R_{r} \right)},R_{r}} \right)}} \\{{= {P_{0} - {\frac{\left( {C_{1} - \frac{R_{r}}{\sigma_{t,r}}} \right){\hat{r}}^{2}}{C_{2}}\left\lbrack {P_{0} + {{\hat{r}}^{2}R_{r}\ln \; \frac{C_{2}}{\left( {C_{1} - \frac{R_{r}}{\sigma_{t,r}}} \right){\hat{r}}^{2}}}} \right\rbrack}}},}\end{matrix} & (19)\end{matrix}$

and the optimal decision by the BS on R_(r) can be obtained by solving

${\frac{\partial{U_{BS}^{*}\left( R_{r} \right)}}{\partial R_{r}} = 0},$

which is the solution R_(r)* to the following equation

$\begin{matrix}{{{\frac{{{f_{1}( \cdot )}{\hat{r}}^{2}} + P_{0}}{\sigma_{t,r}} = {\left( {C_{1} - \frac{R_{r}}{\sigma_{t,r}}} \right){f_{2}( \cdot )}}},{{for}\mspace{14mu} R_{r}},{{{in}\mspace{14mu} {which}\mspace{14mu} {f_{1}( \cdot )}} = {R_{r}{\ln \left\lbrack {g\left( R_{r} \right)} \right\rbrack}}},{{f_{12}( \cdot )} = {\frac{\partial{f_{1}( \cdot )}}{\partial R_{r}}\mspace{14mu} {and}}}}{{g\left( R_{r} \right)} = {\frac{C_{2}}{\left( {C_{1} - \frac{R_{r}}{\sigma_{t,r}}} \right){\hat{r}}^{2}}.}}} & (20)\end{matrix}$

While a closed-form solution for R_(r)* is difficult to derive fromEquation (20), the optimal decision can still be evaluated numerically,given the large computation power of the BS. After obtaining the valueof R_(r)*, the MS will he notified of R_(r)* and the optimal decision ofMS can then be given as

$\begin{matrix}{ɛ^{*} = {\left. {{\overset{\sim}{ɛ}}^{*}\left( R_{r} \right)} \right|_{R_{r} = R_{r}^{*}} = {\sqrt{\ln \; \frac{C_{2}}{\left( {C_{1} - \frac{R_{r}^{*}}{\sigma_{t,r}}} \right){\hat{r}}^{2}}}.}}} & (21)\end{matrix}$

In this interactive decision making process, the BS is responsible formost of the computational task for the optimal decision. For example,the BS first predicts the optimal response function of the MS, and thenderives its own optimal strategy by backward derivation.

In contrast, the MS only needs to determine an optimal value for itsstrategy, instead of a function, after the decision is made at the BS.This imbalance of computational complexity is desirable in practice,because the BS has more advanced computational capacity and is notconstraint in its computational power, while the MS usually operate withlimited computational and battery power.

When

${0 < {\left( {C_{1} - \frac{R_{r}}{\sigma_{t,r}}} \right){\hat{r}}^{2}} < C_{2}},$

the rational outcome of the continuous Stackelberg game is the strategypair {R_(r)*,ε*}, in which the BS promises to reward the additionaltransmission power R_(r)*r_(lo) ² to the MS that performs cognitivesensing with the range of r_(l), while the MS decides to perform sensingwith the range r_(l)*=ε*{circumflex over (r)}. This outcome exists,under the condition of

${0 < {\left( {C_{1} - \frac{R_{r}^{*}}{\sigma_{t,r}}} \right){\hat{r}}^{2}} < C_{2}},$

is the unique Nash Equilibrium for the game. This can be verified fromthe definition of Nash Equilibrium. If the outcome {R_(r)*,ε*} for theBS, given that MS decides to perform sensing with radiusr_(l)*=ε*{circumflex over (r)}, then its utility cannot increase bychanging R_(r)*, because R_(r)* already achieve the maximal possibleutility, per statement (i) and (ii) in FIG. 13. On the other hand, forthe MS, given a reward factor R_(r)*, its utility cannot be increased byselecting a critical value other than ε* either, per statement (iv) inFIG. 13. Because the Nash Equilibrium describes the outcome in which noplayer can be better off by acting independently, the outcome {R_(r)*ε*}is therefore the unique Nash Equilibrium by definition.

Interference Avoidance by Statistical Traffic Models

Inaccuracy of Collected Information Over Time

In a dynamic mobile network, the subcarrier usage information 211 isvalid only for a limited time. However, the information collected cannotbe updated in real time, because continuous updating wastes resources,and is usually impractical. Therefore, periodic updates of current usageare preferred. The length of the interval between the updates cansignificantly impact the performance interference avoidance.

FIG. 14 shows the timing of the update according to an embodiment of theinvention. The duration for the validity of the subcarrier usageinformation is T_(t) 1401, an unknown random variable dictated by theinfrastructure. The BSi is notified of subcarrier usage at a random timet 1404 after the subcarrier has been used for a time T_(a) 1402. Theproblem is to determine the amount of time T_(b) the usage remainsvalid, i.e., T_(b)=T_(t)−T_(a).

Current Age and Future Life of a Current Transmission

We use a renewal-reward process to solve this problem. Renewal-rewardtheory is a branch of probability theory that generalizes Poissonprocesses for random holding times. If the detection occurs at therandom time point t, then the expected duration at the time of detectionis

$\begin{matrix}{{{E\left\lbrack T_{a} \right\rbrack} = \frac{{2{E\left\lbrack T_{t} \right\rbrack}^{2}} - {E\left\lbrack T_{t} \right\rbrack}^{2}}{2{E\left\lbrack T_{t} \right\rbrack}}},} & (22)\end{matrix}$

while the length of future duration T_(b) is

$\begin{matrix}{{{E\left\lbrack T_{b} \right\rbrack} = \frac{E\left\lbrack T_{t}^{2} \right\rbrack}{2{E\left\lbrack T_{t} \right\rbrack}}},} & (23)\end{matrix}$

for any arbitrary distribution of the total duration T_(t).

Constant Traffic Load in the Interfering Cell

We presume that the traffic load in the interfering cell is relativelyconstant over time. This indicates new subcarrier usage is initiatedafter the current subcarrier usage ends. In addition, upon terminationof the current subcarrier usage, a new downlink subcarrier is selectedrandomly after termination. Therefore, the subcarrier usage informationcollected at the time t can be considered accurate for a random durationT_(b)−T_(t)−T_(a).

Information Accuracy and Update Interval

With the periodic update on the current subcarrier usage in the IZ, ifthe period is T_(r), then the information is accurate as long as theallocation of the current subcarrier in the interfering cell does notchange, i.e., the current transmission does not end before the nextround of information collecting. Accordingly, the accuracy of theinformation over time T_(r) is

ξ_(A)(T _(r))=Pr(T _(b) ≧T _(r)).   (24)

Specifically, T_(r)=0 corresponds to the ideal case in which theinformation is collected and updated continuously. We describe theupdating for different types of traffic.

Voice Traffic

Expected Future Life with Exponentially-Distributed Duration

For voice traffic, the duration of each transmission and the subcarrierusage can be described by a light-tail exponential distribution. If theduration of the downlink transmission is denoted by a random variableX=T_(t), then its probability distribution function (pdf) is

$\begin{matrix}{{p_{E}(x)} = \left\{ {\begin{matrix}{{\lambda \; {\exp \left( {{- \lambda}\; x} \right)}},} & {x \geq 0} \\0 & {x < 0}\end{matrix}.} \right.} & (25)\end{matrix}$

Given this distribution on the transmission duration, the expectedfuture life of a current subcarrier usage can be determined, fromEquation (23), as

$\begin{matrix}{{{E\left\lbrack T_{b} \right\rbrack} = {\frac{E\left\lbrack X^{2} \right\rbrack}{2{E\lbrack X\rbrack}} = {\frac{\frac{1}{\lambda^{2}}}{2\; \frac{1}{\lambda}} = \frac{1}{\lambda}}}},} & (26)\end{matrix}$

in which X=T₁ is the random variable that describes the total durationof the voice traffic.

That is, the expected remaining duration of the current subcarrier usagein Γ_(j) is 1/λ, as observed in the cell i at a random time. Thisexpected remaining duration equals the expected duration of T_(t), andcan also be obtained from the unique memoryless property of exponentialdistribution. However, the Equation (22) is more general and is validfor any distribution.

Normalized Information Update Frequency (NIUF)

In practice, the BSi can obtain, through past observation, the rate λ ofthe exponential distribution that governs the duration of the subcarrierusage in the cell j and determine the expected future life of thecurrent transmission. The stations in the cell i then determines thefrequency of update.

A normalized information update frequency (NIUF) is

$\begin{matrix}{{\mu = \frac{E\left\lbrack T_{b} \right\rbrack}{T_{r}}},} & (27)\end{matrix}$

where T_(r) is the interval between two consecutive rounds ofinformation collecting. In this case, the accuracy of the currentInformation over the duration of T_(r) is

$\begin{matrix}{\begin{matrix}{{\xi_{A}\left( {T_{r} = \frac{E\left\lbrack T_{b} \right\rbrack}{\mu}} \right)} = {\Pr \left( {T_{b} \geq \frac{E\left\lbrack T_{b} \right\rbrack}{\mu}} \right)}} \\{= {\int_{\frac{E{\lbrack T_{b}\rbrack}}{\mu}}^{\infty}{{{\lambda exp}\left( {{- \lambda}\; x} \right)}\ {x}}}} \\{= {\exp \left( {{- \frac{E\left\lbrack T_{b} \right\rbrack}{\mu}} \cdot \frac{1}{E\left\lbrack T_{b} \right\rbrack}} \right)}} \\{= {\exp \left( {{- T_{r}} \cdot \frac{1}{E\left\lbrack T_{b} \right\rbrack}} \right)}} \\{{= {\exp \left( {- \frac{1}{\mu}} \right)}},}\end{matrix}{{i.e.},{{\xi_{A}\left( T_{r} \right)} = {{{\exp \left( {{- \lambda}\; T_{r}} \right)}\mspace{14mu} {or}\mspace{14mu} {\xi_{A}(\mu)}} = {{\exp \left( {- \frac{1}{\mu}} \right)}.}}}}} & (28)\end{matrix}$

Single Transition on a Physical Subcarrier During Update Interval

If the current subcarrier usage information is acquired at the time t,and the traffic of the current usage ends at time t+T′, such thatT′<T_(r), then a random subcarrier is assigned to newly-initiatedtraffic, under the assumption of a constant traffic load. We assume inthis embodiment that a specific subcarrier can experience only onetransition during the update interval T_(r).

With the above single-transition assumption, the new traffic does notend before time t+T_(r). The single-transition assumption allows us toinvestigate the impact of inaccuracy over the update interval T_(r),which is not unreasonably long. When studying the performance over thetime T, T can be partitioned into multiple update intervals.

Affects of the Inaccuracy over Time in Blind Optimization

When the information 211 on subcarrier usage in the IZ 105 is exchangedvia the infrastructure 210, it is complete and accurate. The number ofcollisions for blind optimization is described above. However, duringthe interval T_(b), and before the next round of information collectingfor blind optimization, the subcarrier usage in the cell can change, dueto early termination of current transmissions. This renders the currentinformation inaccurate and can introduce additional subcarriercollisions.

The expected number of the additional subcarrier collision that canoccur during a future duration of T, assume T can be partitioned byT_(r), is

$\begin{matrix}\begin{matrix}{{E\left\lbrack {C_{B}^{a}\left( {T,T_{r}} \right)} \right\rbrack} = {\frac{T}{T_{r}}\beta_{t}\beta_{f}{S \cdot \left\lbrack {1 - {\xi_{A}\left( {\mu = \frac{T}{T_{r}}} \right)}} \right\rbrack}{f^{2}(\alpha)}}} \\{= {\frac{T}{T_{r}}\beta_{t}\beta_{f}{S \cdot \left\lbrack {1 - {\exp \left( {- \frac{T_{r}}{T}} \right)}} \right\rbrack}{f^{2}(\alpha)}}} \\{= {{\Phi \left( {T,T_{r}} \right)}.}}\end{matrix} & (29)\end{matrix}$

It can be shown that

${\frac{\partial{\Phi \left( {T,T_{r}} \right)}}{\partial T_{r}} < 0},$

therefore, the expected number of collision introduced by the inaccuracyof the information over time is a monotonic decreasing function ofT_(r), which suggests that shorter updating period, i.e., higher NIUFcan reduce the impact of inaccuracy over time. Further, it can be shownthat

${\lim\limits_{T_{r}\rightarrow 0}{\Phi \left( {T,T_{r}} \right)}} = 0$

for any value of T. The expected number of total subcarrier collisionsover duration of T is

E[C′ _(B)(T,T _(r))]=E[C′ _(B)(T,T _(r))]+E[C _(B) ⁰],   (30)

where E[C_(B) ⁰] equals E[C_(B)] from Equation (3), E[C_(B) ⁰] denotesthe expected number of subcarrier collisions under semi-staticsubcarrier usage, i.e., subcarrier allocation is not dynamic during T,or continuous information update (i.e., T_(r)→0).

By comparison, the expected number of subcarrier collisions with randomsubcarrier allocation, when no information is collected and nointerference avoidance is performed, over a duration of T is

$\begin{matrix}{{E\left\lbrack {C_{R}^{t}(T)} \right\rbrack} = {\frac{T}{T_{r}}{f^{2}(\alpha)}\beta_{i}{\beta_{j}\left\lbrack {2 - {\exp \left( {{- T_{r}}\lambda} \right)}} \right\rbrack}{S.}}} & (31)\end{matrix}$

Both Equations (30) and (31) are based on the single-transitionassumption during the period of T_(r).

It can be shown that when a reasonable update Interval, e.g., whenT_(r)≦E[T_(b)] or μ≧1, is selected, the blind optimization 372 can stilldramatically reduce the expected number of subcarrier collisions, eventhough the subcarrier usage information is inaccurate between the updateintervals. For example, when μ=1, a 80% reduction in subcarriercollision can be achieved when traffic loads is less than about 90%.

Scheduling for Fairness in Time-sharing the Information Update Interval

Upon the collection of the subcarrier usage information, the updatinginterval T_(r) is determined such that the BS can assume with aprediction accuracy ξ_(A)(T_(r)) that currently collected information isvalid for time T_(r). Because such information is usually costly toobtain, careful scheduling is performed to make the best use of theinformation.

Specifically, if multiple MSs in Γ_(i) are to time-share a subcarrierthat is assumed to be collision-free with probability ξ_(A)(T_(r))during the next T_(l) period of time, the fairness of determining thesharing order of the multiple MSs is vital to achieve fairness among theMSs and/or to meet different QoS demands of different MSs.

FIG. 15 shows an example where the information is acquired periodicallyfor intervals T_(r). Current subcarrier usage is detected at the timepoint t. Therefore, no information update on the interferers isavailable during the time interval of [t,t+T_(r)]. The currentsubcarrier x is in use in the cell i. A subsequent newly initiated usageis subcarrier z. Two other MS need downlink transmissions of duration ofT_(A) and T_(r2), respectively, such that T_(r1)+T_(r2)=T_(r). For aneffective usage of the frequency resource, the two MSs can sequentiallyshare a subcarrier y during the next period of time T_(r). Therefore, weneed to determine order for these two MS. The probabilities of collisionfor time slots T_(r1) and T_(r2) are different.

Vulnerability Probability

If the subcarrier y is time-shared by multiple MSs in the cell i duringthe interval T_(r), then P_(k) ^(V) is the probability that the durationof the next newly-initiated subcarrier z usage partially or totallyoverlaps with the transmission duration of the k^(th) MS. In the exampleshown in FIG. 15, the usage of the current subcarrier x ends at timet+T_(b), and when a new subcarrier z can be initiated. BecauseT_(r1)<T_(b)<T_(r1)+T_(r2), the transmission during time T_(r2) isvulnerable to collision, while the transmission during time T_(r1) isnot vulnerable.

Note that when a transmission is vulnerable to collision, it does notmean that a subcarrier collision must happen. Because the newlygenerated traffic in the cell is assigned a subcarrier randomly, the newsubcarrier z may or may not collide with subcarrier y. A collisionoccurs when y=z. The probability of subcarrier collision for the MS incell i that transmits in the k^(th) time slot is P_(k)=P_(k)^(V)·f(α)β_(j).

When two MSs sequentially share the same subcarrier during the timeT_(r), assuming without loss of generality that MS 1 is granted thesubcarrier usage first with T_(r1)=ηT_(r)(0<η<1) and MS 2 is allocatedthe same subcarrier for the rest duration T_(r2)=(1−η)T_(r), thevulnerability probabilities of the two MSs is

$\begin{matrix}\begin{matrix}{{P_{1}^{V}\left( {\eta,\xi_{A}} \right)} = {\Pr \left\{ {T_{b} < {\eta \; T_{r}}} \right\}}} \\{= {\int_{0}^{\eta \; T_{r}}{{{\lambda exp}\left( {{- \lambda}\; x} \right)}\ {x}}}} \\{= {1 - {\exp \left( {{- \lambda}\; \eta \; T_{r}} \right)}}} \\{= {1 - {\xi_{A}^{\eta}\left( T_{r} \right)}}}\end{matrix} & (32) \\{And} & \; \\\begin{matrix}{{P_{2}^{V}\left( {\eta,\xi_{A}} \right)} = {{\Pr \left\{ {T_{b} < T_{r}} \right\}} - {\Pr \left\{ {{T_{b} + T_{t}^{\prime}} > {\eta \; T_{r}}} \right\}}}} \\{= {1 - {\xi_{A}\left( T_{r} \right)} - {\Pr \left\{ {Y_{t} < {\eta \; T_{r}}} \right\}}}} \\{{= {{{\xi_{A}^{\eta}\left( T_{r} \right)}\left\lbrack {1 - {\eta \; \ln \; {\xi_{A}^{\eta}\left( T_{r} \right)}}} \right\rbrack} - {\xi_{A}\left( T_{r} \right)}}},}\end{matrix} & (33)\end{matrix}$

in which Ξ_(A)(T_(r)) is the prediction accuracy associated with timeT_(r) determined in Equation (23), and Y_(t)=T_(b)+T_(t)′ is the sum oftwo exponential random variables. Therefore Y_(t) is a Gammadistribution with a shape parameter 2, and rate λ. It can be shown thatwhen the prediction accuracy is reasonably accurate, e.g.,ξ_(A)(T_(r))>0.2, the vulnerability probability of the first MS isalways lower than that of the second MS. Thus, the first MS is lesslikely to suffer a subcarrier collision.

More generally, if the interval T_(r) is shared among K MSs (K>1), theperiod T_(r) can be partitioned into K time slots and the k^(th) MS isallocated the k^(th) time slot. The duration, of the k^(th) slot isρ_(k)T_(r) such that η₀=0 and

${\sum\limits_{k = 0}^{K}\eta_{k}} = 1.$

The probability of collision in the k^(th) MS is

$\begin{matrix}\begin{matrix}{{P_{k}^{V}\left( \xi_{A} \right)} = {\Pr \left\{ {{\left\lbrack {T_{b},{T_{b} + T_{t}^{\prime}}} \right\rbrack\bigcap\left\lbrack {{\sum\limits_{\gamma = 0}^{k - 1}{\eta_{\gamma}T_{r}}},{\sum\limits_{\gamma = 0}^{k}{\eta_{\gamma}T_{r}}}} \right\rbrack} \neq } \right\}}} \\{= {1 - \begin{pmatrix}{{\Pr \left\{ {T_{b} > {\sum\limits_{\gamma = 0}^{k}{\eta_{\gamma}T_{r}}}} \right\}} +} \\{\Pr \left\{ {\left( {T_{b} + T_{t}^{\prime}} \right) < {\sum\limits_{\gamma = 0}^{k - 1}{\eta_{\gamma}T_{r}}}} \right\}}\end{pmatrix}}} \\{= {1 - {\xi_{A}^{L_{k}}\left( T_{r} \right)} - {F_{Y_{r}}\left( {L_{k - 1}T_{r}} \right)}}} \\{= {{{\xi_{A}^{L_{k - 1}}\left( T_{r} \right)}\left\lbrack {1 - {L_{k - 1}\ln \; {\xi_{A}\left( T_{r} \right)}}} \right\rbrack} - {{\xi_{A}^{L_{k}}\left( T_{r} \right)}.}}}\end{matrix} & (34)\end{matrix}$

Fairness and Prioritization in Scheduling

When multiple MSs in the cell share the transmission interval T_(r)between two consecutive rounds of information update, the analyticalresult: derived above shows that when the MS is allocated to differentslots during T_(r), the probabilities of subcarrier collisions differ.Protocols can be designed to resolve this possible unfairness issue.

Fair scheduling can be implemented by keeping historical record on theslots different MSs are assigned in different intervals. In the examplein FIG. 15, which shows the scenario of two MSs sharing, when T_(r) ischosen such that a reasonable prediction accuracy is achieved and beingassigned the first slot is preferred. If the historical record showsthat MS 1 was usually assigned to less preferred time slot(s), then MS 1should be assigned to the first time slot in current subcarrierallocation, in the interest of fairness.

More generally, for each MS l in the set Γ_(i), let the historical indexbe h=1, 2, . . . , and P_(k) ^(V)(l) be the vulnerability probability ofthe MS l in the current round of subcarrier allocation. The historicalvulnerability probability for the MS l in round h^(th) is recorded asP^(V)(l,h). In this case, when a history of length H(h=0,1 . . . H−1)has been recorded, the current allocation of the MS l to a specifick^(th) slot should be chosen such that

$\begin{matrix}{\left. \frac{{\sum\limits_{h = 0}^{H - 1}{P^{V}\left( {,h} \right)}} + {P_{k}^{V}()}}{H + 1}\rightarrow P_{A} \right.,{\forall{ \in {\Gamma_{i}.}}}} & (35)\end{matrix}$

That is, the current allocation of time slots within the interval T_(r)should be chosen such that the average vulnerability probability overthe historical record approaches P_(A) for all the MSs in the set Γ_(i).

In contrast, if the transmission task of a particular MS has a higherpriority than others, then the BS can exert prioritization over fairnessand allocate the time slot that is associated with the smallestvulnerability probability to the high priority MS. In practice,different criteria can be used to justify higher prioritization such asdelay-sensitiveness, and urgency of the transmission.

Data Traffic

Pareto Distribution and Truncated Pareto Distribution

When the traffic is dominated by data transmission, the distribution ofthe transmission length usually has a heavy-tail, which can be describedby the Pareto or Bradford distribution. The pdf is

$\begin{matrix}{{p_{PE}(x)} = \left\{ {\begin{matrix}{\frac{{ab}^{a}}{x^{a + 1}},} & {x \geq b} \\0 & {x < b}\end{matrix}.} \right.} & (36)\end{matrix}$

When used to describe the duration of data traffic, the Paretodistribution describes a transmission of length at least b and a shapefactor a that usually takes a value from the interval of (0,2), incorrespondence to the heavy-tail property of data traffic. However, arandom variable X governed by the distribution in Equation (36) does nothave well-defined finite moments. Specifically, all finite moments of aPareto distributed X are infinite when 0<a≦1. When 1<a<2, X has a finitefirst moment (mean) but all i^(th)(i>1) are not defined.

In recognition of the analytical unanimity of the unbounded Paretodistribution, a truncated Pareto distribution is usually used todescribe the distribution of the durations of wireless data traffic.Specifically, when there is a higher bound on the maximal length oftransmission m, we use the following truncated Pareto distribution todescribe the distribution of data traffic duration.

$\begin{matrix}{{p_{P}(x)} = \left\{ {\begin{matrix}{\frac{{ab}^{a}}{1 - {\left( \frac{b}{m} \right)^{a}x^{a + 1}}},} & {m \geq x \geq b} \\{0,} & {otherwise}\end{matrix}.} \right.} & (37)\end{matrix}$

In practice, the upper bound in Equation (37) corresponds to the maximalduration of data traffic allowed by the network.

Expected Future Life with Pareto-Distributed Duration

Using the renewal-reward process, the expected future life of thecurrent subcarrier usage for data traffic can be determined fromEquations (23) and (37) as

$\begin{matrix}{{{E\left\lbrack T_{b} \right\rbrack} = {\frac{E\left\lbrack X^{2} \right\rbrack}{2{E\lbrack X\rbrack}} = {\frac{{bm}\left( {a - 1} \right)}{2\left( {a - 2} \right)} \cdot \frac{m^{a - 2} - b^{a - 2}}{m^{a - 1} - b^{a - 1}}}}},} & (38)\end{matrix}$

in which X=T_(l) is the random variable that describes the totalduration of data traffic.

Accuracy of Prediction

Unlike the exponential distribution, the major challenge in assessingthe prediction accuracy for data traffic is that the governing Paretodistribution is not memoryless. Therefore, while the expected value ofthe future life can be determined for the currently-observed subcarrierusage by the renewal-reward process of Equation in (23), assessing theaccuracy of different prediction schemes requires additionalinformation.

If we use NIUF for the data traffic, then the accuracy of the predictioncan be evaluated for different values of the NIUF. When a=1.1, b=2 andm=55, the prediction accuracy for the Pareto distributed data traffic isonly slightly lower than that of exponentially distributed data traffic.

Prediction Based on Current Age

When the history of usage is available, from the infrastructure orcognitive sensing, the accuracy of the currently collected informationover a T_(r) period of time can be evaluated analytically. Specifically,if the usage of a subcarrier is known to have lasted for time T_(a), andthe next round of information collecting is performed at timeT_(r)(T_(r)≦m−T_(a)), then the information is accurate within thisinterval T_(r) with a probability ε_(A)(T_(a),T_(r)), which is

$\begin{matrix}\begin{matrix}{{\xi_{A}\left( {T_{a},T_{r}} \right)} = {\Pr \left( {T_{t} \geq {T_{a} + T_{r}}} \middle| {T_{t} \geq T} \right)}} \\{= \frac{\int_{T_{a} + T_{r}}^{m}{\frac{{ab}^{a}}{1 - {\left( \frac{b}{m} \right)^{a}x^{a + 1}}}\ {x}}}{\int_{T_{a}}^{m}{\frac{{ab}^{a}}{1 - {\left( \frac{b}{m} \right)^{a}x^{a + 1}}}\ {x}}}} \\{= {\frac{\left( {T_{a} + T_{r}} \right)^{- a} - m^{- a}}{T_{a}^{- a} - m^{- a}}.}}\end{matrix} & (39)\end{matrix}$

If the interval between information collecting is T_(r)>m−T_(a), thenthe subcarrier usages in the set Γ_(j) are definitely going to changeduring the interval T_(r), and the prediction accuracy is 0 over thetime interval T_(r). Such a choice is obviously contradictory to thefact that all dada traffic has a length less than m. Therefore, thechoice does not provide any advantage over random allocation over theduration T_(r).

Equivalent to Equation (39), if a particular prediction accuracy ξ_(A) ⁰is desired for the task of interference management and T₀ is available,then the desired accuracy can be achieved, reversely, by determiningthat the interval between information updates is

T _(r) ⁰(T _(a),ξ_(A) ⁰)=[(T _(a) ^(−a) −m ^(−a))ξ_(A) ⁰ +m ^(−a)]^(1/a)−T _(a).   (40)

Determining the Information Collection Interval

Unlike the exponential distribution, the Pareto-distribution has memory.Therefore, different values of different current ages result indifferent value of the expected future validity of usage. However,whether the subcarrier usage information is collected through theinfrastructure or cognitive sensing, it is impractical to collect suchinformation for only one subcarrier each time and perform multiplecollections each round. Instead, the interval between information updateshould be identical for all subcarriers in interest. Then, the BS onlyneeds one round of information collecting for each round of interferencemanagement. This Interval should reflect the expected future validity ofcurrent subcarrier usage, as well as the desired prediction accuracy.

There are two practical approaches that can address this issue. Thefirst one is “average over age” (AoA), and the second is termed as“average over interval” (AoI). For the purpose of determining theinterval between two rounds of information collecting and interferenceavoidance, the current subcarrier usage in the set Γ_(j) is acquired at:the beginning. Meanwhile, for data-traffic, the ages of currently in-usesubcarriers is acquired. Obviously, for each subcarrier s used by a MSin the set Γ_(j), its age will, more often than not, be different fromthat of others, i.e., T_(a,s1)≠T_(a,s2).

In AoA, the mean of the ages of all subcarriers used in the set Γ_(j) isobtained first for the computation of a desired T_(r)*. We assume thatthe subcarrier used in the set Γ_(j) forms the set Ω_(j). Then, theaverage is

${\overset{\_}{T}}_{a} = {\frac{1}{\Omega_{j}}{\sum\limits_{s \in \Omega_{j}}T_{a,s}}}$

and the update interval is determined by Equation (40) as T_(r)*( T_(a),ξ_(A)*).

In AoI, the desired interval for each subcarrier usageT_(r,B)*(T_(a,s),ξ_(A)*) is determined individually for each subcarrierfirst, and the final desired interval is

$\begin{matrix}{T_{r} = {\frac{1}{\Omega_{j}}{\sum\limits_{s \in \Omega_{j}}{{T_{r,s}^{*}\left( {T_{a,s},\xi_{A}^{*}} \right)}.}}}} & (41)\end{matrix}$

The AoI scheme can be evaluated by numerical simulations. If the desiredaccuracy is above 0.5, then the actual prediction accuracy achieved overthe update interval determined by Equation (41) is within 2% of thedesired accuracy.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications may be made within the spirit and scope ofthe invention. For example, though much of this description has focusedon the downlink, the invention is equally applicable to the uplink.Therefore, it is the object of the appended claims to cover all suchvariations and modifications as come within the true spirit and scope ofthe invention.

1. A method for reducing inter-cell interference in a wireless frequencydivision multiplexing network, in which the network includes a set ofbase stations, and in which an area served by each base station forms acell, and in which each cell includes a set of mobile stations served bythe base station, and in which an area of overlap between the cells forman interference zone, and in which a remaining area of the cells notincluding the interference zone form a non-interference zone, and inwhich a spectrum of radio frequencies used by each base station isidentical, and in which the spectrum is partitioned into subcarriers,comprising: allocating the subcarriers randomly to the set of mobilestations in the non-interference zone, in which the allocatedsubcarriers become unavailable, and the remaining subcarriers in thespectrum are available; and selecting in each base station a strategy toallocate the available subcarriers to the set of mobile stations in theinterference zone, in which the strategy is selected from randomallocation, blind optimization, and joint optimization.
 2. The method ofclaim 1, further comprising; allocating the available subcarriers orblocks of subcarriers randomly by each base station if the selectedstrategy is random allocation; allocating the available subcarriers orblocks of subcarriers randomly by one base station and allocating thesubcarriers optimally by an other base station if the selected strategyis blind allocation; and allocating the available subcarriers or blocksof subcarriers optimally by each base station if the selected strategyis joint optimization.
 3. The method of claim 1, further comprising:identifying the set of mobile stations in the interference zone usinghandoff information, in which the handoff information is compared withan interference threshold to determine whether the subcarriers areallocated.
 4. The method of claim 1, further comprising: exchanging ahistorical record of subcarrier usage between the set of base stations.5. The method of claim 2, in which the blind allocation furthercomprises; logically ordering the available subcarriers; and allocatingthe labeled subcarriers in a reverse order by the other base station. 6.The method of claim 2, in which the joint optimization furthercomprises: logically ordering the available subcarriers; allocating thelabeled subcarriers in a forward order by the one base station; andallocating the labeled subcarriers in a reverse order by another basestation.
 7. The method of claim 1, in which, the selecting is performedusing a strategic game.
 8. The method of claim 7, in which the strategicgame uses a sequential decision making process that lead to a Nashequilibrium.
 9. The method of claim 7, in which the strategic game isbase station centric.
 10. The method of claim 7, in which the strategicgame achieves minimum subcarrier collision with a minimum optimizationcost.
 11. The method of claim 7, in which the strategic game is mobilestation centric.
 12. The method of claim 11, in which the strategic gameincludes modeling of a utility function of the mobile station and thebase station to transform power reward into an equivalent overall powersaving.
 13. The method of claim 11, further comprising: collectinginterference information using cognitive sensing, and in which thestrategic game uses a critical sensing radius and a critical ratio ofthe critical sensing radius during the cognitive sensing.
 14. The methodof claim 13, in which an effectiveness of the interference informationcollections is related to the critical sensing radius and the criticalratio.
 15. The method of claim 11, in which the strategic game is abinary Stackelberg leader-follower game.
 16. The method of claim 7, inwhich the strategic game is adaptive to traffic load and a size of theinterference zone.
 17. The method of claim 3, further comprising:identifying the set of mobile stations in the interference zone usingcognitive sensing.
 18. The method of claim 13, further comprising:transmitting probe signals by the set of mobile stations in theinterference zone; and responding subcarrier usage in response toreceiving the probing signals.
 19. The method of claim 18, furthercomprising: rewarding the mobile stations transmitting the probingsignals with increased transmission power from the set of base stations.20. The method of claim 19, in which transmitting and the rewarding usesa strategic game to determine a sensing radius for the probing signal.21. The method of claim 4, in which the allocating is according to thehistorical record of subcarrier usage.
 22. The method of claim 3, inwhich the interference threshold is about ten dB less than a handoffthreshold.
 23. The method of claim 1, further comprising: updatingperiodically current subcarrier usage information in the set of basestations.
 24. The method of claim 23, in a time interval for theupdating depends on types of traffic, and the types of traffic includevoice traffic and data traffic.
 25. The method of claim 23, in which theupdating uses a normalized information update frequency.
 26. The methodof claim 23, in which the allocating of the subcarriers uses fairscheduling.
 27. The method of claim 1, in which set of base stations usea handoff protocol to collect interference information, and theinterference information is used during the allocating and selecting.